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If $\kappa=\lambda^+$ then $|[\alpha]^{<\kappa}|=|[\alpha]^\lambda|$, and $|\alpha^{<\kappa}|=|\kappa\cdot\alpha^\lambda|=|\alpha^\lambda|$, so we reduced this to an equality you already know.

If $\kappa$ is singular, then we can write this as the union of $|[\alpha]^\lambda|$ for a cofinal sequence of $\lambda<\kappa$, and again we reduce this to the previous situation, only now we have two infinite sums over the same index set (the cofinality of $\kappa$), and the summands are the same. Equality ensues.